Lectures on Discrete Geometry (Jiri Matousek)

Errata and remarks

P. 6, the second remark under the separation theorem,
stating that the convex hull of a closed set X equals
the intersection of all closed halfspaces containing X,
is false, since the convex hull of a closed set need not
be closed. (Noted by Uli Wagner.)

P. 7, the proof of the separation theorem (the case of general C and D,
non-strict separation) doesn't work as stated.
(Noted by Wolfgang Weil.)
First of all,
C is not contained in the union of the C_n but only in their closure.
Second, and more serious, the C_n and D_n as defined in the proof need not
be disjoint! (For example, take C as a union of an open halfplane H
and a point p on its boundary, and D as another point q on the boundary
of H, and use p and q as centers of the shrinking.)
To have them disjoint, one needs to choose the center
of the shrinking as a point in the relative interior. This requires
more work, and with this adjustment, this proof is perhaps no longer
the method of choice. Here is an outline of one possible
proof.
First, a standard step for simplyfing matters is noticing that
separation of C from D is equivalent to separating A:=C-D={x-y:
x\in C, y\in D} from {0}. Let B be the closure of A.
If 0 is not in B, we are done. Otherwise, a small argument
shows that 0 must be on the boundary of A, and
so we can choose a sequence {x_n} of points outside B
converging to 0 and verify that separating hyperplanes
of B and x_n
have a convergent subsequence whose limit is a separating
hyperplane (containing 0 and with all of A on one side).

P. 8 bottom, the Bonnice-Klee theorem should read
"Any k-interior point of conv(X) is a k-interior point of conv(Y)
for some Y\subseteq X with at most max(2k,d+1) points,
where x is called a k-interior point of a set C if x lies in the relative
interior of the convex hull of some k+1 affinely independent points
of Y." (Noted by Sergio Cabello.)

P. 9, Exercise 4(b), it is to be proved f(conv(X))=conv(f(X)).
(Noted by Sergio Cabello.)

P. 9, Exercise 6(b), we need to assume that C is a closed
convex cone. (Noted by Yonathan Mizrahi.)

P. 19, in Proposition 2.1.3 we want N and n to be at least 1,
but for m we have to admit m=0 as well. Similarly in Exercise 2.2.5.
(Noted by Marek Scholle.)

P. 21, 2nd line from below: BU should be ZU. (Noted by Sergio Cabello.)

Exercise 2.1.2, we should really have k+1 points instead of k
(with k the result is valid too but the natural statement is with k+1).
(Noted by Leonard Schulman.)

P. 23, the conclusion of the proof of Theorem 2.2.2 is oversimplified;
the point v' need not lie in P. One needs to rewrite
v'=\sum_{j=1}^{i-1}(\alpha_j+1)\gamma_j z_j + \gamma_i\alpha_i w,
and reduce once more; that is, to consider
v'' = \sum_{j=1}^{i-1} fract.part((\alpha_j+1)\gamma_j)z_j
+\gamma_i\alpha_i w,
obtaining \gamma_i=0 as before. (Noted by Sergio Cabello.)

P. 23 (what an unfortunate page!),
Bibliography and Remarks, 3rd line: An
assumption is omitted, namely, that C has to be a star body,
i.e. a compact set containing 0 in the interior and
such that x\in C implies tx\in C for all t\in[0,1].
A few lines below: "det(Lambda) > D" should be "det(Lambda) < D".
(Noted by Sergio Cabello.)

P. 34, the 6-hole problem was resolved in 2005 by Tobias Gerken,
who showed by a clever case analysis that any set containing
9 points in convex positions has a 6-hole, and independently
by Carlos M. Nicolas. Gerken's proof was simplified
by Pavel Valtr to about 4 pages (but one needs more than 9 points in convex
position to start with, and hence the resulting upper bound for n(6) is worse).

P. 44, above the picture, the Fano plane is of order 2, not 3
(noted by Gabriel Nivasch).

P. 46, "Point-plane indicences", the first upper bound
for point-plane incidences is attributed to
Aronov and Agarwal by mistake. The result can be found
in a paper by Guibas, Overmars, and Robert [Comp. Geom. Theor. Appl.
6(1996) 215-230]. (Noted by Yoshio Okamoto.)

P. 55, in the proof of the crossing number lemma we should
assume that no two edgs sharing a vertex cross in the chosen drawing of G,
for otherwise, E[x']=p^4 x need not hold. For straight-line drawings
this assumption is satisfied automatically, and for curvilinear
drawings it is easily enforces.

P. 56, 2nd line below the picture, P should be E. (Noted by Nati Linial.)

P. 131, as I've learned from Ricky Pollack, Oleinik and Petrovskii
only proved bounds for the sum of Betti numbers
(and in particular, for the number
of connected components)
of an algebraic set, i.e. the intersection of the zero
sets of n polynomials, under the additional
assumption that these zero sets are smooth hypersurfaces.
Thom and Milnor obtained more general bounds but still for algebraic
sets. Sign patterns were explicitely considered only later
(but a reduction of this problem to the number of
connected components of a suitable algebraic set is not very hard).

P. 139, Exercise 2(c): There are actually quite simple examples (which I've
overlooked, unfortunately) showing that the number of intersection graphs
of convex sets is at least 2^{const. n^2}. Recently Pach and Toth ("How many drawings
are there?", manuscript, 2002) determined the right constant in the exponent:
the number is 2^{(3/8+o(1))n^2}, for both intersection graphs of planar convex sets
and intersection graphs of planar curves.

In the last displayed formula in the proof of Lemma 6.3.2,
(r/dn) should be (r/2n). (Noted by Yoshio Okamoto.)

P. 158 top picture, there is a missing case (a trapezoid
with a top side, bottom side, left vertical side but
unbounded on the right. (Noted by Gabriel Nivasch.)

P. 159, above the second displayed formula from below,
t(r/n) should be t(n/r).
(Noted by Uli Wagner.)

P. 163, the remark on optimal cuttings for arrangements
of spheres is unfounded (or confusing at best). As far as I could
find, the best known decompositions for arrangements of spheres
need O(n^d log n) cells, and the corresponding (1/r)-cuttings
have r^d log r cells, probably slightly suboptimal. This error of mine,
unfortunately, has been propagated in the (marvellous)
paper of Jozsef Solymosi and Van Ha Vu,
Near optimal bounds for the Erdos
distinct distances problem in high dimensions
[Combinatorica 28,1 (2008): 113-125].

P. 178 middle, definition of "nonrepetitive segment",
the index of the last symbol a_{i+k} should be i+k-1
(noted by Pavel Valtr)

P. 182 Exercise 1, \psi_s(n,m) should be \psi_s(m,n).
(Noted by Yoshio Okamoto.) In part (d), it seems that
the tools given in (a)-(c) do not lead to the claimed
exponent of the logarithm in a simple way.
One can get some suitable exponent c(s) for every s.
(Noted by Gabriel Nivasch.)

P. 203, colored Tverberg theorem: it should be said that the A_i
are (d+1)-element sets. (Noted by Peter Landweber.)

P. 204 4th line of the last paragraph, A should be X.
(Noted by Gabriel Nivasch.)

P. 205 line 2: T_3(X) is always nonempty for a (2d+3)-point set in dimension d.
A correct formulation is "for a set X in R^d, where d is a part of input, is NP-complete."
(Noted by Emo Welzl.)

P. 210, it is incorrectly claimed that EVERY centerpoint
of X is contained in 2/9 of all X-triangles. Boros and Furedi
showed this only for a point of the maximum depth (which may
be larger than n/3 for some X). (Noted by Nabil Mustafa.)

P. 214 above the picture, "3-regular" should be "3-uniform".
(Noted by Yoshio Okamoto.)

P. 218 Lemma 9.3.2: the sets C_i should also be compact,
for otherwise, the implication (i) implies (ii) need not hold.
(Noted by Uli Wagner.)

P. 219, line 9: "no more than 2^{d-1} halvings" should
be "no more than 2^d halvings". (Noted by Gabriel Nivasch.)

P. 224, picture, the second Y_3 should be Y_4.
(Noted by Salman Parsa.)

P. 225 line -8 (after "Therefore,"), the second inequality
is not valid, and so one needs to carry the middle term over
to the subsequent calculation, which thus becomes more complicated.
Here is a fix.
(Noted and correction suggested by Jan Kyncl.)

P. 236 footnote, the known result is weaker than
claimed there: The assumption needed for the hardness
of approximation with a better factor is not P not equal to NP.
It is a stronger but still quite believable assumption about the hardness
of NP-complete problems. (Noted by Yoshio Okamoto.)

P. 237, Exercise 10.1.6(b), all of the inequality signs should be flipped.
(Noted by Leonard Schulman.)

P. 239, Epsilon net theorem: An additional assumption on the measure
\mu is needed; for a pathological \mu the conclusion may fail
(because certain functions in the proof are not measurable).
The theorem is valid, in particular, for every Borel measure
on R^n, and more generally
on every Polish space (separable and metrizable by a complete metric).
This should cover all cases one may normally encounter in discrete
geometry. The precise measurability assumptions needed are stated
in the Vapnik-Chervonenkis paper; one is in the first
paragraph on the page 265 and another is in the middle of page 268 in the
second paragraph of section 3. (Noted by Sergei Starchenko.)

P. 259, Theorem 10.6.1, and next page, Lemma 10.6.2: with
the current argument in the proof of the lemma, we need to assume
that the sets are compact. (Noted by Yoshio Okamoto.)

P. 269, first displayed formula, the sum in the r.h.s. should
be under expectation. (Noted by Gabriel Nivasch.)

P. 275, the line lambda_v in the upper picture should
be drawn one "floor" lower, so that the marked vertices
lie on the middle level. (Noted by Micha Sharir.)
The same adjustment for the lower picture (Noted by Gabriel Nivasch.)

P. 276, end of the 2nd paragraph, the definition of d_i
should be as on the previous page; that is, d_i is the number
of intersections of l_i with l_j, j<i. (Noted by Gabriel Nivasch.)
Later on, instead of
"each line in the bundle of l_i has exactly d_i vertices of V_{m+1}"
it should be
"each line in the bundle of l_i has at most D_m vertices of V_{m+1}".
(Noted by Guenter Rote.)

P. 280, Exercise 1(a), k+1 should be replaced by 2(k+1);
k+1 counts only the directed k-edges going from left to right.
The same correction also applies to the hint to Exercise 2.
(Noted by Gabriel Nivasch.)

P. 286, in the outline of the proof of the improved
k-set bound in dimension 3, n_p should be replaced by n
throughout. Moreover, for the case where the chains C_1 and C_2
do not cross, we still need to distinguish two subcases:
If both C_1 and C_2 end with infinite rays on both sides,
then the argument in the outline can be applied.
The other case, where at least one of C_1, C_2 ends
with at least one bounded edge, needs to be accounted for separately:
there are at most c_p n such pairs. (Noted by Uli Wagner.)

P. 286, 6th line below the first picture, t\in {\cal T}
should be T\in {\cal T}. (Noted by Yoshio Okamoto.)

P. 287, top picture - the point q^* should be on the line pq, rather than
on rq. (Noted by Yoshio Okamoto.)

P. 291, Berge's Strong Perfect Graph Conjecture
was proved by Chudnovsky, Robertson, Seymour, and
Thomas, by graph-theoretic (as opposed to polyhedral) methods.
See
Chvatal's web page about it. Even more recently, a polynomial-time
algorithm for recognizing perfect graphs was found by
Chudnovsky, Cornuejols, Liu, Seymour, and Vuskovic (to appear
in Combinatorica).

P. 298 l. 3, x should be x_1. (Noted by Bernard Lidicky.)

P. 298 bottom, in the "basic fact from measure theory"
one should assume vol(X_1) finite (which is no problem in
the application since the considered sets are compact). (Noted by Eva Kopecka.)

P. 294 Exercise 2: K should be a MAXIMAL clique. (Noted by Robert Samal.)

P. 300, 3rd line from the bottom: A+B should be (1-t)A+tB.
(Noted by Robert Samal.)

P. 301, end of "Bibliography and Remarks", the sets should be
A,B,K_3,K_4,...,K_n and the left-hand side of the inequality
should be V(A,B,K_3,...,K_n)^2. (Noted by Robert Samal.)

P. 301, Exercise 2, A and B should be assumed nonenmpty.
(Noted by Yoshio Okamoto.)

P. 317, the second paragraph ("By more refined consideration...")
is oversimplified and confusing at best. Here is an attempt at a correct
concise explanation: "By more refined consideration, one can improve the lower bound
to approximately the square of the quantity just given.
Having generated the query points x_1,...,x_N for B^n,
we check that for K=conv {+e_1,-e_1,...,+e_n,-e_n,+x_1^0,-x_1^0,...,+x_N^0,
-x_N^0},
with x_i^0=x_i/||x_i||,
the oracle is free to give the same answers as those for B^n,
and the same holds for the dual body K^*.
A deep result (the inverse Blaschke-Santalo inequality)
states that vol(K)vol(K^*) > c^n/n!
for any centrally symmetric n-dimensional convex body K,
with an absolute positive constant c, and together with Theorem 13.2.1
this yields the appropriate lower bound for vol(K^*)/vol(K).
This improvement is interesting because for
symmetric convex bodies it almost matches the performance
of the best known algorithm." (Noted by Nati Linial.)

P. 318, third line from below, "distance at most 1/n from S'" should be "distance at most 1/n from x". (Noted by Man-Cho A. So.)

P. 319, we should (and may) also assume that N exceeds 2n, say,
so that the formula defining k yields a value below n.
(Noted by Robert Samal.)

P. 320, 3rd paragraph of "Bibliography", line 6,
the width of the slab should be 2/||u_i||, rather than
1/||u_i||. (Noted by Yoshio Okamoto.)

P. 323 middle, treatment of the general case,
we need the opposite inequality for q: q=O(n/ln(N/n)).
This is an equally easy calculation. (Noted by
Jeonghyeon Park.)

P. 323, last line, the convex hull of the y^{(i)} is not
a crosspolytope; one should speak about +y^{(i)} and -y^{(i)}
or phrase the argument a bit differently. (Noted by Nati Linial.)

P.324 line 11, "n special points" (instead of "d special points").
(Noted by Nati Linial.)

P. 326, last line, y^{(i)} should be +- y^{(i)}. (Noted by Nati Linial.)

P. 328 Exercise 3: the last inequality in (b)
should be vol(E) <= max(vol(E_1),vol(E_2)), rather than
vol(E) >= min(vol(E_1),vol(E_2)). (Noted by Kasturi Varadarajan.)

P. 329 l. -12, the exponent of epsilon should be +2, rather
than -2.

P. 332 l. 6 (2nd line in the displayed formula), "=" should be
"less or equal" (noted by Rados Radoicic)

P. 333, Prop. 14.2.1, R^d should be R^n. (Noted by Nati Linial.)

P. 339, Prop. 14.3.4, the dimension in the formula should
be one smaller (ie. "-2" instead of "-1"), for otherwise,
the calculation doesn't quite work.
(Noted by Josef Cibulka.)

P. 340, 2nd line of the last paragraph, "-lambda^2"
in the exponent should be "lambda^2".

P. 378 top, phi and eta are interchanged (in the definition,
on the line below the 2nd displayed formula, and 4 lines below).
Moreover, the D^2 factor in the definition of x_{uv} should be
present
in the second case and not in the first, and the inequality
in the next line should be \rho^2(\eta)-D^2\rho^2(\varphi)\geq 0.
Other ways of fixing the confusion with eta and phi are possible as well.
(Noted by Mikhail Ostrovskii.)

P. 383 exercise 15.5.6, the sets A and B should form a partition of V,
i.e., B=V\A. (Noted by Gabriel Katz.)

P. 389, proof of Lemma 15.7.2: r_j should really be
defined as the minimum of r_q and of the current definition.
The argument as written works only for j with r_{j-1}< r_q,
but for the other j we have Delta_j=0. (Noted by Uli Wagner.)